 Competitive equilibrium

Competitive market equilibrium is the traditional concept of economic equilibrium, appropriate for the analysis of commodity markets with flexible prices and many traders, and serving as the benchmark of efficiency in economic analysis. It relies crucially on the assumption of a competitive environment where each trader decides upon a quantity that is so small compared to the total quantity traded in the market that their individual transactions have no influence on the prices. Competitive markets are an ideal, a standard that other market structures are evaluated by.
A competitive equilibrium is a vector of prices and an allocation such that given the prices, each trader by maximizing his objective function (profit, preferences) subject to his technological possibilities and resource constraints plans to trade into his part of the proposed allocation, and such that the prices make all net trades compatible with one another ('clear the market') by equating aggregate supply and demand for the commodities which are traded.
A simple example is a society where there are only 2 products, bananas and apples, and 2 individuals, Jane and Kelvin. The price of bananas is P_{b}, and the price of apples is P_{a}.
The indifference curves J_{1} of Jane and K_{1} of Kelvin first intersect at point X, where Jane has more apples than Kelvin does, Kelvin has more bananas than Jane does, and they are willing to trade with each other at the prices P_{b} and P_{a}. After trading both Jane and Kelvin move to an indifference curve which depicts a higher level of utility, J_{2} and K_{2}. The new indifference curves intersect at point E. The slope of the tangent of both curves equals P_{b}/P_{a}.
And the MRS_{Jane}=P_{b}/P_{a}; MRS_{Kelvin}=P_{b}/P_{a}. The marginal rate of substitution of Jane equals that of Kelvin. Therefore the 2 individuals society reaches Pareto efficiency, where there is no way to make Jane or Kelvin better off without making the other worse off.
The competitive equilibrium and allocative efficiency
At the competitive equilibrium, the value society places on a good is equivalent to the value of the resources given up to produce it (marginal benefit equals marginal cost). By definition, this ensures allocative efficiency (the additional value society places on another unit of the good is equal to what society must give up in resources to produce it)^{[1]}.
Note that microeconomic analysis does NOT assume additive utility nor does it assume any interpersonal utility tradeoffs. Efficiency therefore refers to the absence of Pareto improvements. It does not in any way opine on the fairness of the allocation (in the sense of distributive justice or equity). An 'efficient' equilibrium could be one where one player has all the goods and other players have none (in an extreme example). This is efficient in the sense that one may not be able to find a Pareto improvement  which makes all players (including the one with everything in this case) better off (for a strict Pareto improvement), or not worse off.
References
 ^ Callan, S.J & Thomas, J.M. (2007). 'Modelling the Market Process: A Review of the Basics', Chapter 2 in Environmental Economics and Management: Theory, Politics and Applications, 4th ed., Thompson Southwestern, Mason, OH, USA
See also
Categories: Markets (customer bases)
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